Ends of groups: a nonstandard perspective

We give a nonstandard treatment of the notion of ends of proper geodesic metric spaces. We then apply this nonstandard treatment to Cayley graphs of finitely generated groups and give nonstandard proofs of many of the fundamental results concerning ends of groups. We end with an analogous nonstandard treatment of the ends of relatively Cayley graphs, that is Cayley graphs of cosets of finitely generated groups.


Introduction
Nonstandard analysis made its first serious impact on geometric group theory via the work of van den Dries and Wilkie [7] on Gromov's theorem on polynomial growth. Indeed, the complicated limit used to form the asymptotic cone of a metric space was replaced by an ultrapower, simplifying the proof considerably. More recently, the author used nonstandard methods to settle the local version of Hilbert's fifth problem; see [9].
In this paper, we treat the notion of ends of a finitely generated group from a nonstandard perspective. Roughly speaking, the ends of a topological space are its "path components at infinity." An analsysis of the ends of the Cayley graph of a finitely generated group yields a significant amount of algebraic information about the group. Using the language of nonstandard analysis, the aformentioned heuristic description of the ends of a proper geodesic metric space can be made precise, leading to much simpler and intuitive proofs of many of the fundamental results of the subject.
The aim of this article is two-fold: First, we present nonstandard proofs of several basic facts about the ends of spaces and groups. The idea is to show how the intuitive proofs of these facts can be made into rigorous nonstandard arguments, whence avoiding the lengthy and sometimes messy standard proofs. Ideally, it is my hope that the language and techniques of nonstandard methods can provide simpler proofs of deeper theorems, e.g. Stallings Theorem on groups with infinitely many ends (see Section 5), as well as lead to proofs of new results.
Secondly, we aim to show that certain notions arising naturally in the nonstandard language may lead to classical notions that have yet to be studied. For example, we discuss a nonstandard property that a finitely generated group can possess, namely that the group have multiplicative ends; see Section 6. This notion suggests itself immediately once the nonstandard framework is developed, begging the question of the standard counterpart of the notion. We present several standard characterizations of this notion, one of them being that the group is a semidirect product of a finite group by an infinite cyclic group. It is my belief that there are other such transparent nonstandard notions whose standard counterparts may have yet to be analyzed.
I would like to thank Alberto Delgado for suggesting that I consider ends of groups in a nonstandard way. I would also like to thank Lou van den Dries, Ilya Kapovich, and Patrick Reynolds for useful discussions concerning this work.
This research was conducted while I was a graduate student at the University of Illinois at Urbana-Champaign, where I was supported by a Dissertation Completion Fellowship.

Notations and Conventions
We assume that the reader is familiar with elementary nonstandard analysis; otherwise, consult [6] or [10] for a friendly introduction. Alternatively, [7] contains a short introduction to the subject for group theorists. Here we just fix notations and terminology. To each relevant "basic" set S corresponds functorially a set S * ⊇ S, the nonstandard extension of S. In particular, N and R extend to N * and R * , respectively. Also, any (relevant) relation R and function F on these basic sets extends functorially to a relation R * and function F * on the corresponding nonstandard extensions of these basic sets. For example, the linear ordering < on N extends to a linear ordering < * on N * . Likewise, if G is a group, then the group multiplication m : G × G → G extends to a group operation m * : G * × G * → G * . For the sake of readability we only use a star in denoting the nonstandard extension of a basic set, but drop the star when indicating the nonstandard extension of a relation or function on these basic sets. For example, when x, y ∈ R * we write x + y and x < y rather than x + * y and x < * y. The nonstandard universe is an elementary extension of the standard universe, and when using this fact, we often say that we are using the transfer principle or that we are arguing by transfer.
We remind the reader of the important notion of an internal set. If S and its powerset P(S) are basic sets, then we assume that the membership relation ∈ is a basic relation between S and P(S). Under this assumption, we can canonically identify P(S) * with a subset of P(S * ). After this identification, we call A ⊆ S * internal if it is an element of P(S) * . For internal subsets of N * , there are two important principles that we use frequently throughout the paper, namely overflow and underflow. Overflow states that if A ⊆ N * is internal and contains arbitrarily large elements of N, then A contains an element of N * \ N. Dually, underflow states that if A ⊆ N * is internal and contains arbitrarily small elements of N * \ N, then A contains an element of N.
We also assume that our nonstandard universe is κ-saturated for some sufficiently large cardinal κ (although ℵ 1 -saturation is probably all that is necessary). We remind the reader that this assumption means that whenever (A i | i < κ) is a family of internal sets with the finite intersection property, Throughout this paper, (X, x 0 ) will denote an unbounded pointed metric space. For any point x ∈ X and any R ∈ R, B(x, R) will denote the closed ball centered at x with radius R. For x ∈ X, we let µ X (x) (or simply µ(x) if there is no risk of confusion) denote the monad of x in X, that is the set of points y ∈ X * for which d(x, y) is infinitesimal. We set X ns := x∈X µ(x), the set of nearstandard elements, that is the set of elements of X * which are infinitely close to an element of X. We also set X fin := {x ∈ X * | d(x 0 , x) ∈ R ns }, the set of elements of X * which are within a finite distance to some (equiv. any) element of X. We let X inf := X * \X fin .
When we specialize to the case of groups, we assume that all groups are finitely generated. To avoid trivialities, we also assume that all groups are infinite.

Proper spaces and maps
Recall that a metric space is said to be proper if every closed ball is compact. The following result is well-known, but we include a proof for the sake of completeness.
Lemma 2.1. X is proper if and only if X ns = X fin .
Proof. (⇒) We always have the inclusion X ns ⊆ X fin . Now suppose x ∈ X fin , say d(x, x 0 ) ≤ R with R ∈ R. Since B(x 0 , R) is compact, we have B(x 0 , R) * ⊆ X ns , whence we see that x ∈ X ns .
(⇐) Given any x ∈ X and any R ∈ R, we have B(x, R) * ⊆ X fin = X ns . Thus, given Recall that a map f : X → Y between topological spaces is said to be proper if f −1 (C) ⊆ X is compact for every compact C ⊆ Y .
Proof. (⇒) Suppose f is proper. Fix a basepoint y 0 for Y . Since B(y 0 , n) is compact for every n, there is N n such that d(y 0 , f (x)) > n for every x ∈ X with d(x 0 , x) ≥ N n . Hence, for x ∈ X inf , we have d(y 0 , f (x)) > n for every The following special case of the previous lemma is all we will really need. Recall that a ray in X is just a continuous function r : [0, ∞) → X. Corollary 2.3. Suppose X is proper and r : [0, ∞) → X is a ray. Then r is proper if and only if r(σ) ∈ X inf for every σ ∈ R + inf .

The Space of Ends of a Proper Geodesic Metric Space
In this section, we assume that our unbounded pointed metric space (X, x 0 ) is also proper. We will use the following definition of the ends of a proper metric space.
Definition 3.1. Two proper rays r 1 , r 2 : [0, ∞) → X are said to converge to the same end if for every R ∈ R >0 , there exists N such that r 1 [N, ∞) and r 2 [N, ∞) are contained in the same path component of X \ B(x 0 , R). This defines an equivalence relation on the set of proper rays; the equivalence class of r will be denoted by end(r). The set of equivalence classes will be denoted by Ends(X).
Before we give a nonstandard characterization of two proper rays having the same end, we must introduce and analyze a few nonstandard notions.
Heuristically, one should think of the relation x ∝ y as saying that x and y are in the same "path component at infinity", for there is an internal path connecting x and y which is contained in the infinite portion of the space. Definition 3.3. For x, y ∈ X and R ∈ R >0 , we write x ∝ R y if there is α ∈ C([0, 1], X) such that α(0) = x, α(1) = y, and α(t) ∈ X \ B(x 0 , R) for all t ∈ [0, 1]. Note that ∝ R is an equivalence relation on X for each R ∈ R >0 . We will also use ∝ σ for σ ∈ R * \ R, which is the internal relation on X * given by x ∝ σ y if and only if there is α ∈ C([0, 1], X) * such that α(0) = x, α(1) = y, and α(t) ∈ X * \ B(x 0 , σ) for all t ∈ [0, 1] * . Remark 3.4. Suppsoe x, y ∈ X * . If σ ∈ R * \ R and x ∝ σ y, then x ∝ y. Conversely, if x ∝ y, then, by underflow, there exists ν ∈ N * \ N such that x ∝ ν y; of course such a ν depends on x and y.
Remark 3.5. The choice of [0, 1] * in the above definitions is purely arbitrary. In fact, let Path(X) denote the set of paths in X, that is α ∈ Path(X) if and only if there are r, s ∈ R such that α : [r, s] → X is continuous. Note that any α ∈ Path(X) has a reparameterization in C([0, 1], X). Hence, by transfer, if there is α ∈ Path(X) * , say α : [σ, τ ] → X * , such that α(σ) = x, α(τ ) = y, and α(t) ∈ X inf for all t ∈ [σ, τ ], then x ∝ y; likewise for the notion of ∝ ν . (Here σ and τ are in R * and [σ, τ ] denotes the interval determined by σ and τ in R * .) It follows that ∝ is an equivalence relation on X inf .
In proper geodesic spaces, we can find a "discrete" formulation of ∝. Recall that X is a geodesic metric space if for any x, y ∈ X, there is an isometric embedding α : [0, r] → X such that α(0) = x and α(r) = y, where r := d(x, y). Lemma 3.6. Suppose X is a proper geodesic space. Fix x, y ∈ X * . Then the following are equivalent: (1) x ∝ y; (2) for every ǫ ∈ (R >0 ) * , there is a hyperfinite sequence a 0 , . . . , a ν in X inf such that a 0 = x, a ν = y, and d(a i , a i+1 ) < ǫ for each i < ν; (3) there is a hyperfinite sequence a 0 , . . . , a ν in X inf such that a 0 = x, a ν = y, and d(a i , a i+1 ) ∈ R fin for each i < ν.
(3) ⇒ (1): Let the hyperfinite sequence a 0 , . . . , a ν be as guaranteed to exist by (3). For each i < ν, let [a i , a i+1 ] denote an internal geodesic segment connecting a i and a i+1 . Since d(a i , a i+1 ) ∈ R fin , these segments are contained entirely in X inf . Concatenating these segments and applying Remark 3.5, we see that x ∝ y.
We are now prepared to give a nonstandard characterization of two proper rays having the same end. (1) end(r 1 ) = end(r 2 ); (1) ⇒ (2): Suppose end(r 1 ) = end(r 2 ) and let σ, τ ∈ R >0 inf . Given n, there is N such that for all s, t ∈ R >0 with s, t ≥ N , we have r 1 (s) ∝ n r 2 (t).
The following lemma combines Lemmas 3.6 and 3.7.
For x ∈ X inf , let [x] denote its equivalence class under ∝ and refer to [x] as the infinite path component of x. We denote the set of infinite path components of X by inf . Then Lemma 3.7 allows us to define a map Θ : Ends(X) → IPC(X), Θ(end(r)) = [r(σ)].
Lemma 3.7 further implies that Θ is injective and independent of the choice of σ.
Remark 3.10. The above lemma makes it immediately clear that the proper geodesic space R n , equipped with its usual metric, has two ends if n = 1 and one end if n ≥ 2.
Notation: Let G x 0 (X) denote the set of geodesic rays in X emanating from x 0 .
Proof. Immediate from the proof of Lemma 3.9.
A useful property of a space with finitely many ends is that one can "separate" the ends with a ball centered around x 0 of finite radius. This may not be possible for a space with infinitely many ends. However, we can separate the ends with a ball centered at x 0 of hyperfinite radius. Lemma 3.12. Suppose that X has infinitely many ends. Let Then for every σ ∈ N * \N and for all distinct i, j ∈ I, we have r i (σ) ∝ r j (σ).
If X is a proper geodesic space, then Ends(X) can be topologized in the following manner. Fix r ∈ G x 0 (X). For n > 0, letṼ n (r) be the set of r ′ ∈ G x 0 (X) such that r ′ (m) ∝ n r(m) for all (equiv. some) m > n. Let V n (r) := {end(r ′ ) | r ′ ∈Ṽ n (r)}. Then the sets V n (r) form a neighborhood basis of end(r) in Ends(X).
We now give a nonstandard description of the topology on Ends(X) by giving a description of the monad system of Ends(X). By Corollary 3.11, we can think of Ends(X) as G x 0 (X) modulo the equivalence relation of two rays having the same end. By the Transfer Principle, G x 0 (X) * is the set of internally geodesic rays in X * emanating from x 0 and Ends(X) * is the quotient of G x 0 (X) * modulo the equivalence relation which is the extension of the equivalence relation of two rays having the same end. One should Since r ′ ∈Ṽ n (r) * for each n, we have N ⊆ A. By overflow, there is ν ∈ N * \N such that ν ∈ A. Hence, if σ ∈ R * is such that σ > ν, then r(σ) ∝ r ′ (σ).
inf is such that r ′ (σ) ∝ r(σ). Fix n > 0. We want to show that end(r ′ ) ∈ V n (r) * . For m > n, consider the internal path connecting r ′ (m) and r(m) obtained by first connecting r ′ (m) and r ′ (σ) using r ′ , then connecting r ′ (σ) and r(σ) with an internal path contained entirely in X inf , then finally connecting r(σ) and r(m) using r. This internal path lies entirely in X * \ B(x 0 , n) * , so end(r ′ ) ∈ V n (r) * .
Corollary 3.14. Ends(X) is a compact hausdorff space.
Proof. Lemmas 3.7 and 3.13 make it clear that any two distinct monads are disjoint, whence Ends(X) is hausdorff. Now suppose r ′ ∈ G x 0 (X) * . To show that Ends(X) is compact, we need to find r ∈ G x 0 (X) such that end(r ′ ) ∈ µ(end(r)). The desired geodesic ray is obtained by defining r(t) := st(r ′ (t)); the details are identical to those in the proof of Lemma 3.9.
Equip IPC(X) with the unique topology which makes Θ a homeomorphism. The next lemma gives a more concrete description of this topology on IPC(X).

Ends and Quasi-Isometries
In this section, we use our nonstandard description of ends to give a proof of the fact that quasi-isometries between two proper geodesic spaces induce homemorphisms on the corresponding end spaces. We begin by defining quasi-isometries and proving a few facts concerning quasi-isometries in the nonstandard framework.
We say that f : X → Y is a quasi-isometric embedding if it is a (λ, ǫ)-quasi-isometric embedding for some λ and ǫ. Similarly, f : X → Y is a quasi-isometry if it is a (λ, ǫ)-quasi-isometry for some λ and ǫ. It can be shown that if f : X → Y is a quasi-isometry, then there is a quasi-inverse for f , which is a quasi-isometry g : such that, for all x ∈ X and y ∈ Y , we have d X (g(f (x)), x)) ≤ K and d Y (f (g(y)), y) ≤ K.
Lemma 4.2. Suppose X and Y are proper geodesic spaces and f : X → Y is a quasi-isometric embedding. Then: Proof.
(1) follows immediately from the definition of a quasi-isometric embedding. For (2), fix x, Since g(f (x)) and x are within a finite distance from each other, we have We now have the following well-known standard corollary.
Proof. Suppose, towards a contradiction, that there is n such that for every m, there are x m , x ′ m ∈ X such that x m ∝ m x ′ m and f (x m ) ∝ n f (x ′ m ). Then by saturation, there is x, x ′ ∈ X * such that x ∝ m x ′ for all m and yet f (x) ∝ n f (x ′ ). Set By choice of x and x ′ , A m has the finite intersection property, so by saturation, there is α ∈ m A m . Then α witnesses that x ∝ x ′ . However, f (x) ∝ f (x ′ ), contradicting the previous lemma.
The composition of two quasi-isometries is once again a quasi-isometry. In order to form the quasi-isometry group of X, we first need to identify two quasi-isometries which are a bounded distance away from each other. More precisely, for two functions f, g : X → X, say that f and g are equivalent if there is R ∈ R such that d(f (x), g(x)) ≤ R for all x ∈ X. Then QI(X), the quasi-isometry group of X, will denote the set of equivalence classes of quasi-isometries of X equipped with the operation induced by composition of quasi-isometries.
, from which the continuity of f e follows. Now suppose that g : Y → X is also a quasi-isometry. Then If g happened to be a quasi-inverse to f , then the fact that d (g(f (x)), x) ∈ R fin shows that g(f (x)) ∝ x, whence g e (f e ([x])) = [x] and g e is the inverse to f e . Proof. This is immediate from the previous lemma and Lemma 3.9. The constructions involved show that f e (end(r)) is the end associated to f (r(σ)) for any σ ∈ R >0 inf .

Application to Cayley Graphs of Finitely Generated Groups
In this section, we specialize to the case that X is the metric space associated to the Cayley graph of a finitely generated group. We first consider the more general context of a locally finite combinatorial graph.
Suppose that (V, E) is a locally finite combinatorial graph, that is a combinatorial graph for which every vertex has only finitely many edges emanating from it. We can turn (V, E) into a metric space X := X(V, E) by identifying each edge with an isometric copy of the interval [0, 1] and then declaring, for x, y ∈ X, d(x, y) to be the infimum of the lengths of paths from x to y; see [2] for more details. In this way, X becomes a proper geodesic space. Let us fix a basepoint x 0 of X, which we assume to be an element of V. Let us agree to write V fin for X fin ∩ V * and V inf for V * \ V fin . Since V ∩ B(x 0 , n) is finite for any n, we see that and only if there is a hyperfinite sequence g 0 , . . . , g ν from V inf such that g 0 = v, g ν = v ′ , and (g i , g i+1 ) ∈ E * for all i < ν.
Proof. The backward direction is immediate from the direction (3) ⇒ (1) of Lemma 3.6. For the proof of the forward direction, suppose v ∝ v ′ . By (1) ⇒ (2) of Lemma 3.6, we have a hyperfinite sequence a 0 , . . . , a η in X inf such that a 0 = v, a η = v ′ , and d(a i , a i+1 ) < 1 2 for all i < η. Now define the internal set R ⊆ N * × V * by (i, x) ∈ R if and only if a i and a i+1 lie on the interiors of distinct edges (so in particular, a i , a i+1 / ∈ V * ) and x is the unique vertex lying in between a i and a i+1 . Let π 1 : N * × V * → N * and π 2 : N * × V * → V * denote the projections onto N * and V * respectively. Note that π 2 (R) ⊆ V inf . For j ∈ π 1 (R), let b j ∈ V * be such that (j, b j ) ∈ R. Let η ′ := η+|π 1 (R)| and define a hyperfinite sequence c 0 , . . . , c η ′ from X * ×{0, 1} by internal recursion as follows. Let c 0 = (v, 0). Suppose that i > 0 and that c i−1 has been defined. Then define c i by The idea here is to insert vertices into the original sequence which lie in between consecutive elements of the original sequence. We use 0 and 1 as labels to distinguish original members of the sequence from newly added members of the sequence. Let π : X * × {0, 1} → X * be the projection map. Let η ′′ := |π(X * × {0, 1}) ∩ V * |. Define the hyperfinite sequence d 0 , . . . , d η ′′ from V * by internal recursion as follows. Let d 0 = v = π(c 0 ). Now suppose that i > 0 and d i−1 = π(c j ). Then define d i = π(c k ), where k ∈ {1, . . . , η ′ } is minimal satisfying the requirements that k > j and π(c k ) ∈ V * . Note that successive d i 's are either equal or a distance 1 apart. Let ν := |{d i | i ≤ η ′′ }| and define the hyperfinite sequence g 0 , . . . , g ν by internal recursion as follows. Let g 0 = v = d 0 . Now suppose that i > 0 and g i−1 = d j . Then define g i = d k where k ∈ {1, . . . , η ′′ } is minimal satisfying k > j and d k = d j . This sequence is as desired.
Fix a group G, which by the convention established in the introduction is assumed to be finitely generated and infinite. Fix a finite generating set S for G. We let Cay(G, S), the Cayley graph of G with respect to the generating set S, be the locally finite graph with V = G and edge relation given by (g, h) ∈ E if and only if there is s ∈ S ±1 such that h = gs. (Here, We let X denote the metric space associated to Cay(G, S). We take x 0 = 1 as our basepoint in X. If S ′ is also a finite generating set for G and X ′ is the metric space associated to Cay(G, S ′ ), then X ′ is quasi-isometric to X (see [2], Chapter I.8, Example 8.17(3)), whence Ends(X) and Ends(X ′ ) are homeomorphic by Lemma 4.4. Hence, defining Ends(G) := Ends(X) gives us a space which is uniquely determined up to homeomorphism.
The following group-theoretic interpretation of when g ∝ g ′ follows immediately from Lemma 5.1.
The action of G on itself by left multiplication extends to an isometry of X (as it preserves the relation E), whence Lemma 4.4 yields a group morphism Let H be the kernel of this group morphism, so for h ∈ H and x ∈ G inf , we have hx ∝ x. We will call H the end stabilizer of G. By Lemma 4.2, H is independent of the choice of S. Under the identification between IPC(X) and Ends(X), this morphism becomes g → (end(r) → end(g · r)) : G → Homeo(Ends(X)).
Then for h ∈ H and end(r) ∈ Ends(X), we have end(h · r) = end(r).
We now use everything that we have developed thus far to give a nice nonstandard proof of one of the fundamental theorems of the subject. Theorem 5.3 (Hopf, [11]). Suppose that G has finitely many ends. Then G has at most two ends.
Proof. Suppose that G has finitely many ends but, towards a contradiction, at least 3 ends, say e 1 , e 2 , e 3 . For i = 1, 2, let r i ∈ G 1 (X) be such that end(r i ) = e i . Since H has finite index in G, we see that there is a fixed constant K such that every element of G is within K of an element of H. Thus there is a proper ray r : [0, ∞) → X with end(r) = e 3 and such that |r(n)| ≥ n and r(n) ∈ H for each n. We will need the following claim.
Remark 5.4. There are finitely generated groups with exactly one end. Indeed, the Cayley graph of Z ⊕ Z is quasi-isometric to R 2 , whence Z ⊕ Z has one end. (We will consider a generalization of this fact in Lemma 5.6.) Note that the Cayley graph of Z is quasi-isometric to R, whence Z has two ends. In fact, G has two ends if and only if it is virtually Z, that is if and only if it has a subgroup of finite index which is isomorphic to Z. The "if" direction of this result follows from the fact that if G is a finitely generated group with finite generating set S and G ′ is a finitely generated subgroup of finite index in G with generating set S ′ ⊆ S, then the natural inclusion Cay(G ′ , S ′ ) ֒→ Cay(G, S) of Cayley graphs is a quasi-isometry. The "only if" direction is due to Hopf and will be proved here in Theorem 6.6.
While the proof of Theorem 5.3 given above has the advantage of being rather elementary, we can give an even shorter proof once we establish the following general lemma about the nonstandard extension of the end stabilizer of a group.
Lemma 5.5. Let W ⊆ G inf be internal. Then there is ν ∈ N * \ N such that hx ∈ G inf and hx ∝ x for all x ∈ W and all h ∈ H * with |h| ≤ ν.
Proof. Let A n := {η ∈ N * | η > n}, and for h ∈ H, let For each h ∈ H, we have N ⊆ B h , so the family is a family of internal sets with the finite intersection property, so by saturation, there is γ ∈ n A n ∩ h B h . Consider the internal set Since N ⊆ C, there is ν ∈ N * \ N with ν ∈ C. This ν is as desired.
Here now is a shorter proof of Theorem 5.3. Let x 1 , x 2 ∈ G inf be such that x 1 ∝ x 2 . Fix ν ∈ N * \ N such that hx i ∝ x i for i = 1, 2 and all h ∈ H * with |h| ≤ ν. Fix h ∈ H inf such that |h| ≤ ν and such that h ∝ x 1 and h ∝ x 2 ; this is possible since there is K ∈ N such that every element of G is within a distance of K from an element of H, whence every element of G * is within a distance of K from an element of H * . The proof now proceeds as in the final paragraph of the proof given above. (The fact that |x 1 | = |x 2 | was irrelevant in the proof of Theorem 5.3 given above. Of course, one could take x 1 , x 2 ∈ G inf such that x 1 ∝ x 2 and |x 1 | = |x 2 |.) Let us mention one more application of Lemma 5.2.
Lemma 5.6. If G 1 and G 2 are infinite, finitely generated groups, then G 1 × G 2 has one end.
Proof. We must show that if (g 1 , g 2 ) and (h 1 , h 2 ) are in ( for each i ≤ ν; here S denotes the generating set for G 2 . Then (g 1 , g 2 ) = (h 1 , h 2 ) · (1, s 1 ) · · · (1, s ν ) and each initial segment 2. An identical proof treats the case that g 2 = h 2 ∈ (G 2 ) inf . Now suppose that g 1 ∈ (G 1 ) inf and h 2 ∈ (G 2 ) inf . Then by the special cases just treated above, we have that Remark 5.7. The preceding lemma actually appears in [4] as a corollary of the following more general result: If G contains an infinite, finitely generated normal subgroup H such that G/H is infinite, then G has one end. The proof of this fact is a rather straightforward combinatorial argument, and we were unable to find a nonstandard one simpler than it.
We end this section with a short discussion of amalgamated free products and HNN extensions. This material will be needed in the next section.
(1) Suppose that G 1 and G 2 are groups with subgroups H 1 and H 2 respectively. Further suppose that φ : H 1 → H 2 is an isomorphism. Then the amalgamated free product of G 1 and G 2 with respect to φ is the group A more common notation for this amalgamated free product is where H is a group isomorphic to both H 1 and H 2 . (2) Suppose that G is a group, H 1 and H 2 are subgroups of G, and φ : H 1 → H 2 is an isomorphism. Then the HNN extension of G via φ is the group where t is an element not in G, called the stable letter of G * φ . A more common notation for the HNN extension of G via φ is G * H , where H is a group isomorphic to both H 1 and H 2 .
The following theorem is considered one of the most important theorems in the theory of ends of finitely generated groups. It would be a triumph to find a simple, nonstandard proof of this theorem. Theorem 5.9 (Stallings [15], Bergman [1]). G has more than one end if and only if one of the following holds: where C is a finite group and A = C and B = C, or We end with the Reduced Form Theorems for amalgamated free products (see [14], Theorem 4.1) and HNN extensions (see [3] and [5], Theorem 32). The reduced form theorem for HNN extensions is also referred to as Britton's Lemma.
(1) Suppose that C is a common subgroup of the groups A and B. Then every element g ∈ A * C B can be written in a reduced form where c ∈ C, g 1 , . . . , g n ∈ (A ∪ B) \ C, and for all i ∈ {1, . . . , n − 1}, Moreover, the number n is uniquely determined by g and is called the length of g, denoted ℓ(g).
(2) Suppose that A is a group and φ : C 1 → C 2 is an isomorphism between two subgroups of A. Let t be the stable letter of A * φ . Then every element g ∈ A * φ can be written in a reduced form g = g 0 t ǫ 1 g 1 · · · t ǫn g n , where ǫ i ∈ {−1, 1} for all i ∈ {1, . . . , n}, g i ∈ A for all i ∈ {0, . . . , n}, and there are no subwords of the form t −1 a i t with a i ∈ C 1 or ta i t −1 with a i ∈ C 2 . Moreover, the number n is uniquely determined by g and is called the length of g, denoted ℓ(g).
A nonstandard consequence of this fact is that if G is an amalgamated free product or HNN extension and g ∈ G * is such that ℓ(g) ∈ N * \ N, then g ∈ G inf .

Groups with at Least Two Ends
In this section, we continue to let X denote the metric space associated to Cay(G, S). We further suppose that G has at least two ends. We fix N such that X \ B(1, N ) has at least two unbounded path components. V will always denote the set of vertices of an unbounded path component of X \ B(1, N ). Following Cohen [4], call E ⊆ G almost invariant if the symmetric difference Eg△E is finite for all g ∈ G.
Lemma 6.1. V is almost invariant.
Proof. Fix g ∈ G and h ∈ V inf . Note that hg −1 , hg ∝ h, whence hg −1 , hg ∈ V * . It follows by underflow that for all h ∈ V with |h| sufficiently large, one has that h ∈ V g and hg ∈ V . Fact 6.2. For all but finitely many g ∈ V , we have gV ⊆ V or G \ V ⊆ gV .
Proof. This is actually a special case of Lemma 1.4 from [4], which states that given any two almost invariant subsets E 1 and E 2 of G, then for all but finitely many g ∈ E 1 , one has either gE 2 ⊆ E 1 or G \ E 1 ⊆ gE 2 . (Cohen's Lemma 1.4 has a rather straightforward proof and we have been unable to find a nonstandard proof simpler than his.) Taking E 1 = E 2 = V , which is almost invariant by Lemma 6.1, we see that for almost all g ∈ V , either gV ⊆ V or G \ V ⊆ gV .
Recall that H denotes the end stabilizer of G. Lemma 6.3. For any g ∈ H, we have gV △V is finite.
Proof. Fix g ∈ H and h ∈ V inf . Then since gh, g −1 h ∝ h, we have gh, g −1 h ∈ V * . So by underflow, we have that for h ∈ V with |h| sufficiently large, we have gh, g −1 h ∈ V , finishing the proof. Corollary 6.4. For all but finitely many g ∈ V ∩ H, we have gV ⊆ V .
Proof. By Fact 6.2, for all but finitely many g ∈ V ∩ H, we have gV ⊆ V or G \ V ⊆ gV . However, since G \ V is infinite (as G has at least two ends), the latter alternative contradicts Lemma 6.3. Corollary 6.5. For any g ∈ V inf ∩ H * , one has gV * ⊆ V * .
The proof of the following theorem is essentially the same as in [4], but we include it here for completeness. Theorem 6.6 (Hopf [11], Cohen [4]). If G has at least two ends and has infinite end stabilizer, then G is virtually Z (whence it has exactly two ends). In particular, if G has exactly two ends, then G is virtually Z.
Proof. Fix V as in the beginning of this section. Choose g ∈ V ∩H such that gV ⊆ V ; this is possible by Corollary 6.4 and the fact that H is infinite. Note that then g n ∈ V for all n (whence g has infinite order) and that g −1 / ∈ V . Now note that every x ∈ V can be written as x = g m v, for some m and some v ∈ V \ gV . Indeed, if x ∈ n g n V , then g −n ∈ V x −1 for all n; but Lemma 6.1 tells us that V x −1 differs from V by a finite number of elements of G, yielding a contradiction to the fact that g −n / ∈ V for all n. Likewise, since G \ V is almost invariant, every x ∈ G \ V can be written in the form x = g −m v, for some m and some v ∈ (G \ V ) \ (g −1 (G \ V )). Lemma 6.3 tells us that V \ gV is finite (whence (G \ V ) \ (g −1 (G \ V )) is also finite), and hence the subgroup of G generated by g has finite index in G.
Note that in a group with one end, we have G = H and g ∝ g −1 for every g ∈ G inf = H inf . Contrast this with the following lemma. Lemma 6.7. Suppose that G has two ends. Then for all g ∈ H inf , g ∝ g −1 .
Example 6.9. The free product G := Z 2 * Z 2 is a group with two ends which does not equal its own end stabilizer; here Z 2 denotes the group of two elements. Let a and b be distinct generators for the two factors of Z 2 . To see that G has two ends, notice that reduced words of infinite length are in the same infinite path component if and only if they both begin with a or both begin with b. It then follows that left multiplication by a permutes the two ends of G, so G does not equal its own end stabilizer. Another way to see that G does not equal its own end stabilizer is the observation that any reduced word of infinite length which begins and ends with the same element (e.g. abab · · · a ν factors, ν∈N * \N ) has order 2, whence cannot be in the nonstandard extension of the end stabilizer by Lemma 6.8. Lemma 6.8 leads us to ask what groups G have multiplicative ends: for all infinite g, g ′ ∈ G * , if g ∝ g ′ , then gg ′ ∈ G inf and gg ′ ∝ g? It turns out that there is a standard characterization of groups with this property. We first provide a well-known consequence of Stalling's Theorem for which we were unable to find a reference. The outline of the proof was communicated to me by Ilya Kapovich. Recall that a group G is a (internal) semidirect product of K by Q if K and Q are subgroups of G, K is normal in G, G = KQ, and Lemma 6.10. A finitely generated group G has two ends if and only if G is a semidirect product of a finite group by a group which is isomorphic to Z or Z 2 * Z 2 .
Proof. The "if" direction follows from Example 6.9 and the fact that a virtually two-ended group is itself two-ended. We now prove the "only if" direction. By Stallings theorem, G admits a simplicial cocompact action on a simplicial line T with finite-edge stabilizers. We thus obtain a homorphism α : G → Aut(T ) with finite kernel K. We claim that α(G) is isomorphic to Z or Z 2 * Z 2 .
loss of generality, we suppose that C is a common subgroup of A and B. Fix a ∈ A \ C and b ∈ B \ C. Fix ν ∈ N * \ N. Let g := abab · · · a ν factors . By the reduced form theorem for amalgamated free products, we have that that the elements gbg −1 and gb −1 g −1 are both infinite and in the same infinite path component as g. Hence, gbg −1 ∝ (gbg −1 ) −1 , contradicting (3). Now consider the case of the HNN extension G = A * φ , where φ : C 1 → C 2 is an isomorphism between two subgroups of A. Let t be the stable letter of G. Fix a ∈ A \ C 1 and ν ∈ N * \ N. Then by Britton's Lemma, the elements t −ν at ν and t −ν a −1 t ν are both infinite and in the same infinite path component as t −ν , yielding t −ν at ν ∝ (t −ν at ν ) −1 , contradicting (3).
(1) ⇒ (4) If G were isomorphic to a semidirect product of a finite group by Z 2 * Z 2 , then G has infinite elements of order 2 (see the argument of Example 6.9), whence G does not equal its own end stabilizer by Lemma 6.8. By Lemma 6.10, G must be isomorphic to a semidirect product of a finite group by Z.
(4) ⇒ (5): Suppose G is a semidirect product of the finite group K by an infinite cyclic group L with generator l. Since G is virtually Z, we know that G has two ends. Since conjugation by l is an automorphism of K and K is finite, there must be n such that l n kl −n = k for all k ∈ K. It follows that l n is central in G (and has infinite order).
(5) ⇒ (1): Suppose that G has two ends and has an infinite cyclic central subgroup L generated by l. Suppose, towards a contradiction, that G is not equal to its own end stabilizer. Choose x ∈ G such that x permutes the two ends of G. Fix ν ∈ N * \ N. Then xl ν = l ν x ∝ l ν , contradicting the fact that x permutes the ends of G.
As is well-known, semidirect products are sensitive to the order of the factors; the next lemma exemplifies this fact. Lemma 6.12. If G is a semidirect product of an infinite cyclic group L by a finite group K, then G is a finitely generated group with two ends which is not equal to its own end stabilizer unless G is the direct product of L and K.
Proof. If G is not isomorphic to the direct product of L and K, there must be k ∈ K such that for every l ∈ L, klk −1 = l −1 . Let l ∈ Z inf be arbitrary. Then (lk) · (lk) = l(klk −1 )k 2 = k 2 ∈ G, whence G does not have multiplicative ends, and hence, by Lemma 6.11, G is not equal to its own end stabilizer. Now we consider the situation when G has infinitely many ends, whence the end stabilizer H of G is finite. Proof. Suppose g ∈ G is such that Hg fixes the ends of IPC(X). Fix g ′ ∈ G inf . Then by hypothesis, we have Hg ′ ∝ Hgg ′ , so there are s 0 , . . . , s ν ∈S such that Hgg ′ = Hg ′ s 0 · · · s ν and satisfying Hg ′ s 0 · · · s i ∈ (G/H) inf for every i ∈ {0, . . . , ν}. Write gg ′ = hg ′ s 0 · · · s ν , where h ∈ H. It now follows that gg ′ = hg ′ s 0 · · · s ν ∝ g ′ s 0 · · · s ν ∝ g ′ . Since g ′ ∈ G inf was arbitrary, we have that g ∈ H, completing the proof of the lemma. Corollary 6.14. If G has infinitely many ends, then Ends(G) is homeomorphic to Ends(G ′ ), where G ′ is a group with trivial end stabilizer.
Proof. Since H is a finite normal subgroup of G, Cay(G) and Cay(G/H) are quasi-isometric, whence Ends (G) and Ends(G/H) are homeomorphic. Take G ′ = G/H.
To summarize, if G is a group with at least two ends, then G has infinite end stabilizer if and only if G has exactly two ends. If G has infinitely many ends, then we know that its end stabilizer must be finite, and then in this case, G is quasi-isometric with a group with trivial end stabilizer.

Relative Ends
In this section, we still assume that G is an infinite, finitely generated group with finite generating set S. We further suppose that K is a subgroup of G of infinite index in G. We let Cay(G, K, S), the relative Cayley graph of G with respect to K and S, be the locally finite graph with V = G/K, the set of right cosets of K in G, and such that (Kg, Kg ′ ) ∈ E if there is s ∈ S ±1 such that Kg ′ = Kgs. We let X denote the metric space obtained from Cay(G, K, S). As in the case of the ordinary Cayley graph, if S ′ is also a finite generating set for G and X ′ is the metric space obtained from Cay(G, K, S ′ ), then X and X ′ are quasi-isometric, whence we can speak of Ends(G, K) and IPC(G, K) as the spaces of ends and infinite path components of any relative Cayley graph of G with respect to K.
Since G ∩ K * = K, the natural map ι : G/K → (G/K) * = G * /K * , ι(Kg) = K * g, is injective. Note that for g ∈ G * , K * g ∈ (G/K) fin if and only if there are s 1 , . . . , s n ∈ S ±1 such that K * g = K * s 1 · · · s n , that is (G/K) fin = ι(G/K). In other words, K * g ∈ (G/K) fin if and only if there is x ∈ G such that gx ∈ K * . This leads to the following definitions.
Note that G ⊆ G fin,K and G fin,K = G (whence G inf,K = G inf ) if and only if K is finite. These definitions were made so that the identities (G/K) fin = {K * g | g ∈ G fin,K } and (G/K) inf = {K * g | g ∈ G inf,K } would hold tautologically.
The following is immediate from Lemma 5.1.
Note that the relation ∝ K is an equivalence relation, whence we can speak of the K-infinite path components of G * . Note that g ∝ K g ′ implies that g ∝ g ′ , and if K is finite, then the notion ∝ K is just the notion ∝. The definitions were made so that the following lemma would be a tautology.
Lemma 7.5. For g, g ′ ∈ G inf,K , we have K * g ∝ K * g ′ if and only if there exists h ∈ K * such that g ∝ K hg ′ .
Let N G (K) := {g ∈ G | ghg −1 ∈ K for all h ∈ K} be the normalizer of K in G. Notice that N G (K)/K acts on G/K by left multiplication and this action preserves the relation E, whence we can extend this action to an isometry of X. By Lemma 4.4, we obtain a group homomorphism Kg → ([K * x] → [K * gx]) : N G (K)/K → Homeo(IPC(X)).
Let L be the normal subgroup of N G (K) such that L/K is the kernel of the above morphism, so for l ∈ L and K * x ∈ (G/K) inf , we have K * lx ∝ K * x.